In classical mechanics, the ‘natural’ state of a body is to be at rest or, if in motion, to continue in a straight line at constant speed indefinitely. This is Newton’s 1st Law. Any deviation from this state is to be attributed to the action of a force and, for Newton, all forces were external since Newton knew nothing of chemical bonding or nuclear reactions.
Similarly, since Newton did not have at his disposal the notion of the field (which was only invented by Faraday in the 19th century), he assumed that all forces were to be attributed to the action of other bodies : there were contact forces (such as those due to collisions) or distant forces such as those due to gravitational attraction. And for anything to happen at all one needed at least two bodies and the situation was supposed to be symmetric : if A affects B, then B affects A to exactly the same extent ─ Newton’s 3rd Law.
The important point here is that, in the Newtonian scheme, no body is an island but always “part of the main” (to paraphrase Donne) and this ‘main’, because attraction was universal and instantaneous, turns out to be the entire universe. In the Newtonian schema, every atom is enmeshed in a complex net of forces stretching out in every direction and from which there is no hope of escape (Note 1)

Need for an ‘Inertial Frame’

So, could, in the Newtonian scheme, an entirely isolated body be said to have any properties at all apart from occupying a certain position in space? As Bishop Berkeley observed at the time (Note 2), to speak of a completely isolated body being in ‘motion’ or ‘having a velocity’ is meaningless : we require at least one other body with which we can compare the first body’s changes of position over time. And, in like manner, the ‘second’ body requires the first.
But a ‘two-body system’ where each body is moving relative to the other is not much of an advance on a single body if we want to work out the successive positions of either, or both, of these bodies, especially if they are circling round each other. What is required is a rigid framework which encloses our ‘test’ body and which does not itself move around appreciably while the ‘test’ body is free to move inside it. Hence the idea of an ‘inertial frame’ : an absolutely indispensable concept without which physics would never have developed very far.

Celestial and Terrestrial Inertial Frames

On the astronomical scale, the required framework was supposed to be provided by the ‘fixed stars’ ─ even though it was already known by Newton’s time that the stars were not completely fixed in their positions relative to each other. But compared to the Earth the stars provided a good enough backdrop.
What of terrestrial frames? Galileo’s ‘inertial frame’ was the windowless cabin of a ship conceived to be either at rest or moving at a constant speed on a calm sea (presumably rowed by galley-slaves). Today, we have much better ‘inertial frames’, cars, trains, ocean liners, aircraft, spaceships and so on ─ indeed it is remarkable that Galileo and his contemporaries were able to conceive of the idea of an ‘inertial frame’ at all since methods of transport at the time were so jerky.
Of course, not all physical objects are situated inside recognizable ‘inertial frames’, but, if need be, we simply imagine a frame, usually either the classic Cartesian box frame or a spherical ‘frame’ like that of an idealized Earth (without flattening at the poles). If we take one corner of the box as the fixed origin, or the centre of the Earth, we can fix the position of any small body relative to the ‘origin’ using at most three ‘specifications’, i.e. co-ordinates.
Galileo was not particularly interested in inertial frames as such and only introduced the windowless cabin ‘thought experiment’ to meet the standard objection to the heliocentric theory, “If the Earth is moving round the Sun, why do we not register this movement?” Galileo replied, in effect, that neither do we necessarily register motion here on Earth provided this motion is more or less constant and in a straight line. He challenges a traveller, shut up in the windowless cabin of a ship, to decide whether the ship is at the dock or travelling at constant speed over a calm sea. Galileo argues that no experiment undertaken inside the cabin, would enable the voyager to come to a final decision on the matter. We ourselves know how difficult it sometimes is, when in a train for example, to decide without looking out of the window whether we are in motion (relative to the platform) or are still at the station.
This question of distinguishing between inertial frames has had enormous importance in the history of physics since it ultimately gave rise to Einstein’s Theory of Special Relativity. 19th century physicists, although accepting that no mechanical experiment would be able to distinguish between Galileo’s two situations, reasoned that there ought nonetheless to be a foolproof method of distinguishing between rest and constant straight-line motion by way of optical experiments. The Michelsen-Morley experiment was designed to detect the (very nearly) constant straight-line motion of the Earth through the all-pervading ether. The famous null result caused a crisis in theoretical physics which was only resolved by Einstein. He made it an axiom (assumption) of his theory that no experiment that ever would or could distinguish, from the inside, between different inertial frames. (More precisely, what Einstein assumed was that “the laws of physics take the same form in all inertial frames”. If identical bodies in similar physical conditions were observed to behave differently in different inertial frames, then this would show either that Einstein’s assumption was wrong or that there were no universally valid ‘laws of physics’.

Absolute Motion and Absolute Rest?

Newton himself was reluctant to accept what Galileo’s Principle of Relativity implied : namely that there was no such thing as ‘absolute’ motion, or for that matter ‘absolute’ rest, only motion or rest relative to some agreed body or point in space. Instinctively Newton felt that there ought to be some way of distinguishing between ‘absolute’ and ‘relative’ motion, and consequently between constant straight line motion and rest. But he conceded that, practically speaking, he could not see how this could be achieved ─ “the parts of space cannot be seen or distinguished from one another by our senses, therefore in their stead we use sensible measures of them” (Principia Motte’s translation p. 8).
Newton did, however, point out that we can make an ‘absolute’ distinction when speaking of rotational movement — “There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion” (Newton, Principia p. 11) As evidence for this Newton used the ‘bucket and rope’ experiment.
If a bucket of water is suspended on the end of a twisted rope and we leave the rope to untwist, the water climbs up the sides of the bucket which it would not otherwise do. This was, to Newton, an example of ‘absolute rotational movement’ within an ‘absolute’ frame ultimately provided by the fixed stars. In this case, the situation was not symmetrical : one could, by observing the torsion of the rope, conclude that it was the bucket, and not the stars, that was rotating (Note 4).
In much the same manner, one might well expect there to be some inertial frame that was ‘truly at rest’ and against which the motions of all other inertial frames could be judged.

Abandon of the Principle of Relativity in UET

What is the comparable situation in Ultimate Event Theory? After agonising over this question and related issues for the best part of a year, I have finally taken the plunge and decided to discard one of the most firmly established and fruitful principles in the whole of physical science. So there we are : the Rubicon is crossed.
In UET, the equivalents (sic) of inertial frames are not generally equivalent. In principle at least, it should be possible to distinguish between a ‘truly stationary’ event-chain and a ‘non-stationary’ one, as also between event-chains which have different constant displacement rates. Indeed, I aim to propose an axiom which in effect says just this. At present, contemporary experimental methods most likely do not allow one to make such fine distinctions, but this situation may change during this century, and indeed I predict that it will.
Dispensing with the Principle of Special Relativity does not mean we have to abandon all the formulae and predictions based upon it. There exists now a substantial body of evidence that ‘verify’ the formulae Einstein originally deduced on the basis of his particular assumptions, and in the last resort this evidence is the justification of the formulae, not the other way round. It is, for example, a matter of empirical fact that it is not possible to accelerate a body beyond a certain well-defined limit, and that the closer one gets to this limit, the more difficult it is to accelerate the body.

Assumption of Continued Existence

In ‘classical’ mechanics and physics generally, it is taken for granted that, once in existence, a ‘solid body’ carries on existing more or less in the same shape and form. Even rocks and mountains get worn down in the end, of course, but their constituent ‘bodies’, namely their atoms, last far longer. For Newton and his contemporaries atoms were indestructible, just as they were for the originators of the atomic theory, Democritus and Epicurus. Although twentieth-century discoveries have overturned this rash assumption ─ most elementary particles are very short-lived indeed ─ there remain plenty of small ‘bodies’ all around and inside us, that we are assured have been in existence for millions, sometimes even billions, of years. The idea that “once in existence an object tends to carry on existing indefinitely” is so deeply ingrained in Western thought that it has rarely been seriously questioned.
In Western thought but not Eastern. Two of the principal Indian and Chinese systems, Buddhism and Taoism, on the contrary emphasize the transience and ephemerality of all physical (and mental) phenomena. According to Buddhism nothing lasts for more than an instant and even solid objects such as rocks and corpuscles are flickering in and out of existence even as we look at them. It is significant that the ancient Chinese equivalent of the (under normal conditions) unchanging elements in the Periodic Table are the shifting configurations of the Y Ching, the Book of Changes.
Now, in UET the ‘natural’ state is for every ultimate event to appear and disappear for ever. If an ultimate event reappears and keeps on doing so, thus inaugurating an event-chain, this can only be due to a ‘force’ ─ I have thought of calling it ‘existence-force’. Most ultimate events never become subject to ‘existence force’ ─ never acquire ‘existence energy’ if you like ─ but, once they do acquire this capacity to repeat, they generally retain it for a considerable length of time. Once an event-chain is established, then, no extra force, inner or outer, is required for it to ‘keep on existing’ : on the contrary effort is required to terminate an event-chain, i.e. to stop the ultimate event or event-cluster repeating. And this is a very important fact.

Why is acceleration so difficult?

Why is it so difficult to make a particular object ‘go faster’? And why, the faster an object is already ‘moving’, is it all the more difficult to make it go faster still?
This state of affairs might appear ‘obvious’, but I do not believe that it is. Take a bath with a little water in it. Does it become more difficult to add one more teaspoonful as the bath fills up? No, it does not. And even when the bath is full, you can still merrily carry on adding water, though in this case some of the water will spill onto the floor. On the other hand, it is extremely difficult to get a lot of personal objects to fit into a travelling case : we have to fold clothes carefully so that they lie flat, arrange solid objects so they fit together neatly and so on.
What is the difference between the two sets of examples ─ the bath and the travelling case or trunk? It is, of course, simply a matter of the available space. In the case of an ‘open’ container, such as a bath, there is more or less unlimited space; in the case of a trunk, the available space is seriously limited.
Now apply this to ultimate events (which are the equivalent of ‘elementary particles’ in UET). Taking as our starting point the spot where an event occurs at a particular ksana, there is seemingly a built in limitation to how far away the next event in the event-chain can occur. If there is to be only one ‘next event’, it can only occur in a single spatial direction relative to its predecessor ─ as opposed to all three directions at once. And there is an upper limit to the possible ‘lateral distance’ between successive events if they are members of an event-chain. This is so because the ‘range’ of the causal connection is finite ─ everything in UET except the Event Locality itself is finite. There may conceivably be relations of some sort between two events that are separated by more than c emplacements at successive ksanas, but, in UET as in the theory of Special Relativity, such relations cannot be causal, at any rate as the term is normally understood (Note 5).
In UET everything is static though one static set-up is constantly being replaced by another. ‘Motion’ in UET simply means the replacement of one ultimate event or event-cluster by another event or event-cluster. Instead of particles in perpetual motion, we must think rather in terms of evanescent point-like ultimate events encased in ‘event containers’. In the proposed schema for UET, each ultimate event has its own particular ‘event-capsule’ of variable dimensions. If we label the boundary positions in any one spatial direction 0 and c , we can say that there are c* = (c – 1) possible emplacements for ultimate events inside a single capsule in a single direction. (This excludes the two boundary positions.) But only one of these positions or ‘event-pits’ can be occupied at a single ksana (moment).

…………………….………………..●

0 ← c* → c

Now, there is seemingly, also a limit on how far the very next ultimate event in an event-chain can be displaced in a single direction. This is a matter of experience and observation though it would be difficult to imagine a ‘world’ in which there was not a limit of some sort. If there was no such limit, something that I do here, wherever ‘here’ is, could have immediate consequences at some arbitrarily distant spot in the universe. The speed of the transmission of causality would be ‘infinite’. This is scarcely conceivable and, in any case, for the purposes of UET, one can simply rule out any such possibility by invoking the ‘Anti-Infinity Postulate’. Eddington, rightly in my view, argued that one could decide for strictly a priori reasons that there must be a ‘speed limit’ for the transmission of energy (or information) in any universe, though one could not for a priori reasons decide exactly what this limit must be.
There is, then, a permanent constraint on all event-chains without exception : successive ultimate events cannot be more than c* positions apart in any one direction. And if we have two event-chains where the distance between successive events of each of the two chains regularly increases by, say, d positions (where d < c*) at every ksana, there is a further constraint on this dual system, namely that the greatest possible subsequent increase is (c* – d) emplacements. (Note that I am speaking of event positions or emplacements not distances in the ‘metric’ sense.)
In effect, looming over and above each individual ‘event-capsule’ with its ultimate event, there is a sort of ghostly potential event-container which dictates how far the next ultimate event of an event-chain can be relative to its previous position. If we label the boundary positions of this ‘macroscopic’ event-container 0 and c , we can say that this creature has the capacity to accommodate c* ultimate events in any one spatial direction but no more. It is, in effect, a scaled-up version of an individual event-capsule since, in the case of an individual event-capsule, there are exactly the same number of possible emplacements for an ultimate event ─ but only one position can be occupied at a time. This parallelism turns out to be extremely significant in UET.
In matter-based physics, we say that a ‘body’ cannot go any faster than c metres per second. The equivalent statement in UET would be : “It is not possible to fit more than c* ultimate events into a causal event container”. Once this container is ‘full’, there is no room for any more events and that is that. This question of available space, and the increasing difficulty of cramming events into it, is the crucial issue in UET from which all sorts of consequences follow. As this available space becomes curtailed, the system as a whole becomes subject to increasing pressure and strongly resists any further constriction. To speak in mathematical terms, any supposed ‘event-packing function’ ─ the equivalent of the acceleration function ─ would not be linear, would start off almost as a straight line but would rise precipitously as v gets nearer and nearer the maximum possible value c* = (c – 1).

The Inertial Ratchet

This picture of an event-container and ultimate events inside it is, of course, not quite right. If we are considering an event-chain where each constituent event is ‘laterally displaced’ at each successive ksana, all the intermediate possible emplacements ─ spots where ultimate events could in principle have occurred ─ are not actually occupied. But it is as if they were. There is no way of going back to the previous state of affairs ─ except by applying a completely new force. Galileo’s notion of inertia should not be interpreted negatively, i.e. as showing our personal incapacity to distinguish ‘inertial frames’, but realistically as a sort of ‘valve’ or ‘space-time ratchet’ which stops an event-chain reverting to its previous occurrence pattern. Not only can the ‘moving finger’ of Omar Khayyam not be lured back to “cancel half a line” but it must inexorably keep on writing at the same rate. If, then, for some reason, an event-chain A suddenly increases its lateral distance from event-chain B by d emplacements at every ksana, it must seemingly keep increasing its distance by this precise amount of d event-emplacements indefinitely.
This property of maintaining a constant ‘speed’ without extra effort is an astonishing and extremely important fact about physical reality which has been glossed over because of the exclusive concentration on the technicalities of how one might actually be able to distinguish between one ‘inertial frame’ and another. Galileo, foreshadowed by the great medieval thinker Oresme, realized that it is not the distance between two bodies (event-clusters) that is important, but the increase in the distance. Why should this be? Because, as far as we know, there is no built in restriction on how far two event-chains can be apart. But the doctrine of the equivalence of all inertial frames means that conditions within any one of the inertial frames remain exactly the same whether or not the two frames (repeating event-clusters) are right alongside each other or are moving apart at a fantastic speed provided this speed is constant ksana by ksana (and less than the upper limit).
But can one really believe this? One can ─ or I can ─ understand only too well why people (including Newton) were disinclined to accept Galileo’s Principle of Relativity and subsequently at first even more disinclined to accept Einstein’s more extended version. Only repeated experiments of increasing precision made some such acceptance mandatory.

The Systems Axiom

Let us examine the reasons for this reluctance. It has been argued that velocity has little if any meaning if we are speaking of a completely isolated body, aka event-chain. We thus require at least two bodies that then form a dual system. And although, if we are confined to the point of view of one ‘inertial frame’ (which we naturally consider to be at rest), we will attribute a certain ‘velocity’ to the other inertial frame (if it seems to be getting further away at each successive moment), this ‘velocity’ really belongs to the dual system ─ and not to either of the components of the system to the exclusion of the other. Very well. Considering the dual system, can we say that a situation where this system is expanding by d emplacements per ksana is ‘equivalent’ to a situation where it is expanding by 2d emplacements per ksana, or for that matter by zero emplacements per ksana? Or even by c* emplacements per ksana? Clearly, the situations or configurations of the dual system are not equivalent, cannot possibly be. However, we are asked to accept that the situations in each of the two components of the system are indistinguishable. Is this reasonable? No.
Why is it not reasonable? Because one would expect conditions of the system as a whole to have repercussions of some sort on all parts of the system : indeed, it is hardly conceivable that it could be otherwise. Certainly in most physical contexts this is what we find. If two bodies are linked by gravitational attraction, this systems situation is detectable in either one of the two (or more) bodies ─ provided we have sensitive enough instruments, of course. Similarly, for components of an electrical circuit. Indeed, one could argue that Newton’s 3rd Law makes something of the kind not just possible but obligatory.
This can be presented as an axiom :

If a system as a whole is subject to certain constraints, then so are all parts of the system and in a similar manner.

Another statement of the principle would be :

If two or more configurations of a dual event-chain system are distinguishable when considering the system as a whole, then the configurations of each of the two or more components of the system must also be distinguishable considered individually.

Now, this axiom is incompatible with the Principle of Relativity, or at any rate what the Principle is taken to imply, namely that there is no way, from the inside, of distinguishing between inertial frames. Let us take a practical example.

The spaceship on the way to the moon

Once a spaceship bound for the Moon has got sufficiently outside the Earth’s gravitational grip, the rocket motors are turned off. Neglecting minuscule perturbations from other planets, comets and so on, the rocket carries on at the same speed relative to the Earth and in more or less exactly the same direction : it does not drop back to what it was before. To reproduce previous conditions, say when the rocket was momentarily stationary relative to the Earth, it would be necessary to start other motors firing, i.e. to introduce a new force.
Now, so we are assured, astronauts devoid of radio contact, windows and so on, would not be able to tell whether the rocket or spaceship was ‘in motion’ relative to the Earth or motionless. The dual system ‘rocket/Earth’ is very definitely not the same in the two contexts. And the same goes for Galileo’s ‘port/ship’ system. If the ship is being rowed by galley-slaves on a calm sea, the distance between the repeating event-clusters we call ‘port’ and those we call ‘ship’ is increasing at every ksana, and so is the distance between the rocket on its way to the Moon and the Earth.
If now we apply the ‘Systems under Constraint Axiom’, we must conclude that there is, at least in principle, some way of distinguishing between the two situations. Why so? Because the constraints on the system are not the same. In the case of a stationary dual system, there is the constraint that, at the subsequent ksana, the distance between the two components can be at most c* units. If the system is already expanding by d units of distance, then there is the more stringent constraint that no increase greater than c* – d units of distance is possible. And in the case when the system is expanding at the maximum possible rate, c* positions per ksana, no further increase is possible at all : the constraint becomes a total ban.
So, according to the ‘Systems under Constraint Axiom’, since the system as a whole is under constraint, each component of the dual system is also under constraint and this constraint should in principle be observable. How can this be? Well, first of all, I need to work out a schema that allows such a distinction to be observable in principle; subsequently, it is for experiments to detect such a distinction, or some eventual consequence of such a distinction.
This is why, as stated earlier, I have eventually come to the unwelcome conclusion that the schemas of Ultimate Event Theory and Relativity diverge : they are not ‘homologous’ as mathematicians might put it. This topic will be pursued in subsequent posts. SH 1/8/14

Note 1 One is reminded at once of the human (pseudo)individual enmeshed in the web of karma. Except that, according to the Buddha, there is hope of escape.

Note 2 “Up, down, right’ left, all directions and places are based on some relation and it is necessary to suppose another body distant from the moving one ….. so that motion is relative in its nature, [and] it cannot be understood until the bodies are given in relation to which it [a particular body] exists, or generally there cannot be any relation, if there are no terms to be related. Therefore, if we suppose that everything is annihilated except for one globe, it would be impossible to imagine any movement of that globe.”Bishop Berkeley, quoted Sciama

Note 3 “All things are placed in time as to order of succession and in space as to order of succession. It is from their essence or nature that they are places; and that the primary places of things should be movable is absurd. (…) But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures for them. For from the positions and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions….. And so, instead of absolute places and motions, we use relative ones.” Newton, Principia ‘Scholium’ p.8 Motte’s translation

Note 4 The late 19th century Austrian physicist Mach argued that the two descriptions, Earth rotating, Heavens fixed and Heavens rotating, Earth fixed, were equally valid and this was also Einstein’s view.

Note 5 Entangled photons and other particles do give rise to event correlations that far exceed c, but asuch associations of distant events are not considered to be causal in the normal sense. The issue bothered Einstein so much that he never accepted Quantum Mechanics in its then current form, nor would he have been any happier with it in its present form.